cp's OEIS Frontend

A253298 Digital root for the following sequences, F(4*n)/F(4); F(12*n)/F(12); F(20*n)/F(20), where the pattern increases by 8, ad infinitum, with the Fibonacci numbers F = A000045.

%I A253298 #63 Jul 10 2016 23:45:43
%S A253298 1,7,3,5,5,3,7,1,9,8,2,6,4,4,6,2,8,9,1,7,3,5,5,3,7,1,9,8,2,6,4,4,6,2,
%T A253298 8,9,1,7,3,5,5,3,7,1,9,8,2,6,4,4,6,2,8,9,1,7,3,5,5,3,7,1,9,8,2,6,4,4,
%U A253298 6,2,8,9
%N A253298 Digital root for the following sequences, F(4*n)/F(4); F(12*n)/F(12); F(20*n)/F(20), where the pattern increases by 8, ad infinitum, with the Fibonacci numbers F = A000045.
%C A253298 Cyclical and palindromic in two parts with periodicity 18: {1, 7, 3, 5, 5, 3, 7, 1, 9, 8, 2, 6, 4, 4, 6, 2, 8, 9}.
%C A253298 Digital root of the period is 9, its mean and median is 5, and its product is (9!)^2.
%C A253298 See A253368 for the initial motivation for this sequence.
%C A253298 From _Peter M. Chema_, Jul 04 2016: (Start)
%C A253298 A composite of three respective digital root sequences in alternation: a "halving sequence" of 1, 5, 7, 8, 4, 2, a "doubling sequence" of 7, 5, 1, 2, 4, 8, and a three-six-nine circuit of 3, 3, 9, 6, 6, 9.
%C A253298 Also the digital root of A000045(4n)/3 or A004187(n).  In general terms, sequences defined by Fib(x*n)/ Fib(x) where x=(8*a-4) all share the same digital root (e.g., F(4*n)/F(4); F(12*n)/F(12); F(20*n)/F(20); F(28*n)/F(28); F(36*n)/F(36), etc.) (End)
%H A253298 Tom Barnett, <a href="https://www.youtube.com/watch?v=kxuU8jYkA1k">Phi VBM Tori Array</a>, YouTube video (see first two minutes).
%H A253298 <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1).
%F A253298 a(n) = A010888(A253368(n)).
%F A253298 G.f.: x*(1 + 7*x + 3*x^2 + 5*x^3 + 5*x^4 + 3*x^5 + 7*x^6 + x^7 + 9*x^8 + 8*x^9 + 2*x^10 + 6*x^11 + 4*x^12 + 4*x^13 + 6*x^14 + 2*x^15 + 8*x^16 + 9*x^17)/(1 - x^18). - _Vincenzo Librandi_, Mar 28 2016
%t A253298 f[n_] := Mod[ Fibonacci[ 12n]/144, 9]; Array[f, 5*18] (* _Robert G. Wilson v_, Jan 23 2015 *)
%t A253298 LinearRecurrence[{1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1},{1, 7, 3, 5, 5, 3, 7, 1, 9, 8, 2, 6, 4, 4, 6, 2, 8},72] (* _Ray Chandler_, Aug 12 2015 *)
%Y A253298 Cf. A010888, A253368, A070366 and A001370.
%K A253298 nonn,easy
%O A253298 1,2
%A A253298 _Peter M. Chema_, Dec 30 2014
%E A253298 Edited. Numbers and name changed to fit A253368. Formula adapted. Cross reference added. - _Wolfdieter Lang_, Jan 28 2015
%E A253298 Name generalized by _Peter M. Chema_, Jul 04 2016